Harmonic Sequence

We must first understand what arithmetic sequence or arithmetic progression is before learning about harmonic sequence or harmonic series. An arithmetic series is a set of numbers where the difference between any two consecutive elements is always constant. Arithmetic Progression, Geometric Progression, and Harmonic Progression are three forms of progression.

In this article, we are going to discuss the harmonic sequence ,harmonic progression in maths and its formula along with solved examples. 

Harmonic Sequence Definition

The harmonic sequence in mathematics can be defined as the reciprocal of the arithmetic sequence with numbers  other than 0. The sum of harmonic sequences is known as harmonic series. It is an infinite series that never converges to a limit.

For example, let’s take an arithmetic sequence as 5, 10, 15, 20, 25,... with the common difference of 5.

Then its harmonic sequence is: 1/5, 1/10, 1/15,1/20,1/25….

Harmonic Progression in Maths

In Mathematics, we can define progression as a series of numbers arranged in a predictable pattern. It's a form of number set that adheres to strict, predetermined laws, which is the main difference between a progression and a sequence, as a sequence is solely based on specific logical rules.

A Harmonic Progression (H.P.) is a series of real numbers that can be determined by multiplying the reciprocals of an arithmetic progression that doesn't contain zero. For example, for the arithmetic progression, p, q, r, s,... 

The harmonic progression is: 1/p, 1/q, 1/r, 1/s,...

What does Harmonic Mean?

The inverse of the arithmetic mean of the reciprocals is used to measure the harmonic mean. So, if 1/a and 1/b are two consecutive terms, their harmonic mean is given by : 

\[ H = \frac{2ab}{a+b} \]

The following is the formula for calculating the harmonic mean for n terms:

Harmonic Mean = \[ \frac{n}{ [(\frac{1}{a})+(\frac{1}{b})+(\frac{1}{c})+(\frac{1}{d})+...] } \]

Where a, b, c, d are the values and n is the total number of values present.

First-Term of Harmonic Progression 

The first term of the harmonic progression is denoted by a. The sum of the series can never be an integer except for the first term, as it can be 1.

Common Difference of Harmonic Progression

The common difference is the difference between any two consecutive numbers in the series. The common difference is denoted as ‘d’ and it is the same in any progression. 

Example of First Term and Common Difference of H.P.

If 1/a, 1/b, 1/c are three terms in Harmonic Progression, then the first term is 1/a and the common difference is d.

So, the common difference will be,

\[ d = \frac{1}{a} - \frac{1}{b} = \frac{1}{c} - \frac{1}{b} \]

or \[ \frac{a-b}{ab} = \frac{b-c}{bc} \]

or \[ \frac{a}{c} = \frac{b-c}{a-b} \]

Harmonic Progression Formula

In an H.P with n terms, we need the formula to find the value of its n^th term. This formula is equal to the reciprocal of the formula for finding the n^th term of arithmetic progression.

Thus, the formula to find the n^th term of the harmonic progression series is given below:

n^th term of the Harmonic Progression = \[ \frac{1}{[a + (n−1)]} \times d \]

Where

“a” is the first term of H.P.

“d” is the common difference

“n” is the number of terms in H.P.

Therefore, we can say that the n^th term of H.P =  \[\frac{1}{(n^{th} \text{ term of the corresponding A.P)}}\]

Sum of Harmonic Progression 

The sum of n terms in a harmonic progression can be determined easily if the first term and the value of n terms is known. 

If the terms \[ \frac{1}{a}, \frac{1}{a+d} , \frac{1}{a+2d}..., \frac{1}{a+(n-1)d} \] make a harmonic progression, the formula to find the sum of n terms in the harmonic progression can be obtained by the formula: 

Sum of n terms, \[ S_{n} = \frac{1}{d}\] In \[ \frac{2a+(2n-1)d}{2a-d} \]

Where,

“a” is the first term of H.P.

“d” is the common difference of H.P.

“ln” is the natural logarithm

Properties of Harmonic Progression

• No term of H.P. can be zero.

• If H is the H.M. between a and b, then we have the following properties:

\[ \frac{1}{H-a} + \frac{1}{H-b} = \frac{1}{a} + \frac{1}{b} \]

(H - 2a)(H - 2b) = H²

\[ \frac{H+a}{H-a} + \frac{H+b}{H-b} = 2 \]

• If a and b be two positive real numbers, then the relation between A.M, G.M and H.M is: A.M x H.M = G.M²

Uses of Harmonic Sequence in Real Life

Some important applications of harmonic progression in everyday life include:

1. Scientists use harmonic formulas to determine the result of their experiments.

2. Harmonic sequencing is used in the field of music to practice notes.

3. It is also used in electrical gadgets or machines and the generation of power.

4. The formula of harmonic progression is applied to determine the degree at which water boils when its temperature is modified by the same value each time.

5. Businesses and large corporations also employ harmonic progression to efficiently run their functions such as sale predictions, financial budgeting, Weather Forecasting, etc.

Did You Know?

The harmonic sequence was first studied back in the 6th century by the Greek mathematician Pythagoras. He first used harmonic progression to study the nature of the universe and also to study music.

Also, a trick to solve harmonic progression questions easily is to convert H.P into arithmetic progression whenever possible.

Solved Examples

1. Find the sum of the harmonic sequence: 1/12 + 1/24 + 1/36 +1/48 +1/60

Solution: As we know, the reciprocal of H.P terms form an A.P

So, for the given H.P, the A.P terms will be 12, 24, 36, 48, 60

Here a = 12, d = 24 - 12 = 12, n = 5

Now, put the above values in the sum of A.P formula

\[ S_{n} = \frac{n}{2} {2a+(n-1)d} \]

So put the value of a,d and n in the above equation

\[ S_{5} = \frac{5}{2} (2\times 12 +(5-1)12) \]

\[ = \frac{5}{2}(24+48) \]

= 180

We know, the sum of H.P will be the reciprocal of the sum of A.P

So, the sum of 5 terms of the given H.P will be \[\frac{1}{180}\]